Approximate joint block diagonalization (AJBD) of a set of matrices has applications in blind source separation, e.g., when the signal mixtures contain mutually independent subspaces of dimension higher than one. The main message of this paper is that certain ordinary approximate joint diagonalization (AJD) methods (which were originally derived for degenerate" subspaces of dimension 1) can als...

A recently developed Thouless-expansion-based diagonalization-free approach for improving the efficiency of self-consistent field (SCF) methods (Noga and Šimunek, J. Chem. Theory Comput. 2010, 6, 2706) has been adapted to the four-component relativistic scheme and implemented within the program package ReSpect. In addition to the implementation, the method has been thoroughly analyzed, particul...

Results are presented which show precise ways in which recursion rests on very simple computational bases which do not support diagonalization. A method based on recursion and making no use of diagonalization is given for proving lower bounds on computational complexity. Thus the intractability of computational problems such as Presburger arithmetic does not depend on diagonalization. This rese...

Precoding with block diagonalization is an attractive scheme for approaching sum capacity in multiuser multiple input multiple output (MIMO) broadcast channels. This method requires either global channel state information at every receiver or an additional training phase, which demands additional system planning. In this paper we propose a lattice based multi-user precoder that uses block diago...

We discuss a method called quasi-sparse eigenvector diagonalization which finds the most important basis vectors of the low energy eigenstates of a quantum Hamiltonian. It can operate using any basis, either orthogonal or non-orthogonal, and any sparse Hamiltonian, either Hermitian, non-Hermitian, finite-dimensional, or infinite-dimensional. The method is part of a new computational approach wh...

We present a set of gradient based orthogonal and nonorthogonal matrix joint diagonalization algorithms. Our approach is to use the geometry of matrix Lie groups to develop continuous-time flows for joint diagonalization and derive their discretized versions. We employ the developed methods to construct a class of Independent Component Analysis (ICA) algorithms based on non-orthogonal joint dia...

A common question from students on the usual diagonalization proof for the uncountability of the set of real numbers is: when a representation of real numbers, such as the decimal expansions of real numbers, allows us to use the diagonalization argument to prove that the set of real numbers is uncountable, why can't we similarly apply the diagonalization argument to rational numbers in the same...

In Blind Source Separation problems it is assumed that the approximate diagonalization of a matrix set achieves more robust solutions than the simultaneous diagonalization of a matrix pencil. In this work we will analyse approximate diagonalization methods using a generalized eigendecomposition (GED) of any pair of a given matrix set. The constraints of GED solutions provide a criterion to choo...

The stability and sensitivity of joint diagonalization criteria are analyzed using the Hessian of the criteria at their critical points. The sensitivity of some known joint diagonalization criteria is shown to be weak when they are applied to the matrices with closely placed eigenvalues. In such a situation, a large deviation in the diagonalizer cause a slight change in the criterion which make...

Reverberant signals recorded by multiple microphones can be described as sums of sources convolved with different parameters. Blind source separation of this unknown linear system can be transformed to a set of instantaneous mixtures for every frequency band. In each frequency band, we may use the simultaneous diagonalization algorithms to separate the sources. In addition to our previous simul...